Showing papers on "Cancellative semigroup published in 2012"

On maximal subgroups of free idempotent generated semigroups

Abstract: We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite semigroup. (3) Every group is a maximal subgroup of some free regular idempotent generated semigroup. (4) Every finite group is a maximal subgroup of some free regular idempotent generated semigroup arising from a finite regular semigroup. As a technical prerequisite for these results we establish a general presentation for the maximal subgroups based on a Reidemeister-Schreier type rewriting.

Inverse semigroup C*-algebras associated with left cancellative semigroups

Abstract: To each discrete left cancellative semigroup $S$ one may associate a certain inverse semigroup $I_l(S)$, often called the left inverse hull of $S$. We show how the full and the reduced C*-algebras of $I_l(S)$ are related to the full and reduced semigroup C*-algebras for $S$ recently introduced by Xin Li, and give conditions ensuring that these algebras are isomorphic. Our picture provides an enhanced understanding of Li's algebras.

A Sufficient Condition for the Non-Finite Basis Property of Semigroups

Abstract: Recently, Zhang and Luo proved that a certain semigroup L of order six is non-finitely based. The main aim of the present article is to generalize this result to a sufficient condition for the non-finite basis property of semigroups. It follows that the semigroup L is inherently non-finitely based relative to a certain class of semigroups. It is also shown that the variety var L generated by L contains a unique maximal subvariety that is non-finitely based. Consequently, the variety var L is not a limit variety.

Aczélian n-ary semigroups

Abstract: We show that real continuous, symmetric, and cancellative n-ary semigroups are topologically order-isomorphic to additive real n-ary semigroups. The binary case (n=2) was originally proved by Aczel (Bull. Soc. Math. Fr. 76:59–64, 1949); there symmetry was redundant.

On the growth of generating sets for direct powers of semigroups

Abstract: For a semigroup S its d-sequence is d(S) = (d1, d2, d3, . . .), where di is the smallest number of elements needed to generate the ith direct power of S. In this paper we present a number of facts concerning the type of growth d(S) can have when S is an infinite semigroup, comparing them with the corresponding known facts for infinite groups, and also for finite groups and semigroups.

On the K-theory of the C*-algebra generated by the left regular representation of an Ore semigroup

Abstract: We compute the K-theory of C*-algebras generated by the left regular representation of left Ore semigroups satisfying certain regularity conditions Our result describes the K-theory of these semigroup C*-algebras in terms of the K-theory for the reduced group C*-algebras of certain groups which are typically easier to handle Then we apply our result to specific semigroups from algebraic number theory

On the Betti numbers of some semigroup rings

Abstract: For any numerical semigroup S, there are infinitely many numerical symmetric semigroups T such that S = T/2 (see below for the definition of T/2) is their half. We are studying the Betti numbers of the numerical semigroup ring K[T ] when S is a 3-generated numerical semigroup or telescopic. We also consider 4-generated symmetric semigroups and the so called 4-irreducible numerical semigroups.

On Ordered Ternary Semigroups

Abstract: . We introduce the concepts of ordered quasi-ideals, ordered bi-ideals in anordered ternary semigroup and study their properties. Also regular ordered ternarysemigroup is de ned and several ideal-theoretical characterizations of the regular orderedternary semigroups are furnished. 1. IntroductionThe literature of a ternary algebraic system was introduced by D. H. Lehmer[3] in 1932. He investigated certain ternary algebraic systems called triplexes whichturn out to be ternary groups. The notion of ternary semigroup was known toS.Banach. He showed by an example that ternary semigroup does not necessar-ily reduce to an ordinary semigroup. In [6] M. L. Santiago developed the theory ofternary semigroups. He focused his attention mainly to the study of regular ternarysemigroups, bi-ideals and ideals in ternary semigroups. The semigroup Z of all in-tegers under multiplication which plays a vital role in the literature of semigroup.The subset Z + of all positive integers of Z is a semigroup under multiplication.Now if we consider the subset Z of all negative integers of Z, then it is not a semi-group under multiplication. Taking these facts in mind D. H. Lehmer [3] introducedthe notion of ternary semigroup. Z is a natural example of a ternary semigroupunder the ternary multiplication. N. Kehayopulu in [5] developed the theory ofpo-semigroups. He mainly studied regular po-semigroups, ideals and bi-ideals inpo-semigroups. In 1999, Sang Keun Lee and Seong Gon Kang [4] gave charac-

Decomposition of Semigroup Algebras

Abstract: Let A⊆B be cancellative abelian semigroups, and let R be an integral domain. We show that the semigroup ring R[B] can be decomposed, as an R[A]-module, into a direct sum of R[A]-submodules of the quotient ring of R[A]. In the case of a finite extension of positive affine semigroup rings, we obtain an algorithm computing the decomposition. When R[A] is a polynomial ring over a field, we explain how to compute many ring-theoretic properties of R[B] in terms of this decomposition. In particular, we obtain a fast algorithm to compute the Castelnuovo–Mumford regularity of homogeneous semigroup rings. As an application we confirm the Eisenbud–Goto conjecture in a range of new cases. Our algorithms are implemented in the Macaulay2 package MonomialAlgebras.

Proper covers of restriction semigroups and w-products

Abstract: Each factor semigroup of a free restriction (ample) semigroup over a congruence contained in the least cancellative congruence is proved to be embeddable into a W-product of a semilattice by a monoid. Consequently, it is established that each restriction semigroup has a proper (ample) cover embeddable into such a W-product.

Fixed point theorems for a semigroup of generalized asymptotically nonexpansive mappings in CAT(0) spaces

Abstract: In this paper, we prove the existence of common fixed points for a generalized asymptotically nonexpansive semigroup { T s : s ∈ S } Open image in new window in CAT(0) spaces, when S is a left reversible semitopological semigroup. We also prove Δ- and strong convergence of such a semigroup when S is a right reversible semitopological semigroup. Our results improve and extend the corresponding results existing in the literature.

Bipartite graphs and completely 0-simple semigroups

Abstract: In a manner similar to the construction of the fundamental group of a connected graph, this article introduces the construction of a fundamental semigroup associated with a bipartite graph. This semigroup is a 0-direct union of idempotent generated completely 0-simple semigroups. The maximal nonzero subgroups are the corresponding fundamental groups of the connected components. Adding labelled edges to the graph leads to a more general completely 0-simple semigroup. The basic properties of such semigroups are examined and they are shown to have certain universal properties as illustrated by the fact that the free completely simple semigroup on n generators and its idempotent generated subsemigroup appear as special cases.

Torsion and Tensor Products Over Domains and Specializations to Semigroup Rings

Abstract: Let R be a commutative Noetherian domain, and let M and N be finitely generated R-modules. We give new criteria for determining when M tensor N has torsion. We also give constructive formulas for producing a module in the isomorphism class of the torsion submodule of M tensor N. In some cases we determine bounds on the length and minimal number of generators of this module. We focus on the case where R is a numerical semigroup ring with the goal of making progress on the Huneke-Wiegand Conjecture.

On the transition semigroups of centrally labeled rauzy graphs

Abstract: Rauzy graphs of subshifts are endowed with an automaton structure. For Sturmian subshifts, it is shown that its transition semigroup is the syntactic semigroup of the language recognized by the automaton. An inverse limit of the partial semigroups of nonzero regular elements of their transition semigroups is described. If the subshift is minimal, then this inverse limit is isomorphic, as a partial semigroup, to the -class associated to it in the free pro-aperiodic semigroup.

Ornstein-Uhlenbeck theory in finite dimension

Abstract: The contents of these notes were presented during ten lectures, in November 2011, by Peter Sjogren in Gothenburg. The text was written by Adam Andersson who participated and is improved after the careful reading by Peter Sjogren. Ornstein-Uhlenbeck theory can be described as a model of harmonic analysis in which Lebesgue measure is everywhere replaced by a Gaussian measure. The theory has applications in quantum physics and probability theory. If one passes to infinite dimensions and places the theory in a probabilistic context, one gets the Malliavin calculus. In Chapter 1, the basic theory is developed. This concerns the Hermite polynomials, the Ornstein-Uhlenbeck operator and most importantly its semigroup. The Hermite polynomials form an orthogonal system with respect to the Gaussian measure in Euclidean space. It turns out that they are the eigenfunctions of the Ornstein-Uhlenbeck operator, and since this operator is self-adjoint and positive semidefinite, the semigroup can be defined spectrally. An explicit kernel is derived for the semigroup, known as the Mehler kernel. It will be of central importance in this text. In Chapter 2, boundary convergence for the semigroup is considered, i.e., the limiting behavior of the semigroup as the “time” tends to zero. This is done by introducing a maximal operator for the semigroup and proving that it is of weak type (1,1). This result implies almost everywhere convergence for integrable boundary functions. In Chapter 3, first-order Riesz operators related to the Ornstein- Uhlenbeck operator are treated. Explicit off-diagonal kernels for these operators are found. It is finally proved that the Riesz operators are of weak type (1,1).

On Intra-Regular Ordered Ternary Semigroups

Abstract: In this paper we give some characterizations of the intra-regular or-dered ternary semigroups in terms of bi-ideals and quasi-ideals, bi-idealsand left ideals, bi-ideals and right ideals of ordered ternary semigroups. Mathematics Subject Classification: 06F05, 06D72, 08A72, 20N99,06F99Keywords: Ordered ternary semigroup, right (left, bi-, quasi- ) ideal,ideal, intra-regular ordered ternary semigroup 1 Introduction In 1932, Lehmer gave the definition of ternary semigroups. A nonempty set T is calleda ternary semigroup ifthereexistsa ternaryoperation T×T×T → T ,written as ( a,b,c ) −→ abc satisfying the following identity ( ∀a,b,c,d,e ∈T )((( abc ) de )=( a ( bcd ) e )=( ab ( cde )).Any semigroup can be reduced to a ternary semigroup but a ternary semi-group does not necessarily reduce to a semigroup. However, Banach showsthat a ternary semigroup does not necessarily reduce to a semigroup by thisexample.Example 1.1 ([4]) T = {−i, 0 ,i} is a ternary semigroup while T is not asemigroup under the multiplication over complex numbers.

Quotient semigroups and extension semigroups

Abstract: We discuss properties of quotient semigroup of abelian semigroup from the viewpoint of C*-algebra and apply them to a survey of extension semigroups. Certain interrelations among some equivalence relations of extensions are also considered.

Split quasi-adequate semigroups

Abstract: The so-called split IC quasi-adequate semigroups are in the class of idempotent-connected quasi-adequate semigroups. It is proved that an IC quasi-adequate semigroup is split if and only if it has an adequate transversal. The structure of such semigroup whose band of idempotents is regular will be particularly investigated. Our obtained results enrich those results given by McAlister and Blyth on split orthodox semigroups.

A cayley theorem for the multiplicative semigroup of a field

Abstract: Since there exist two commutative elementarily equivalent semigroups of which one is the multiplicative semigroup of a field and the other is not a multiplicative semigroup of any field, it is impossible to characterize multiplicative semigroups of fields by formulas of the first order language (logic). In this work we characterize the multiplicative semigroup of a field by its binary representation (Cayley type theorem).

On some semigroups of the partial transformation semigroup

Abstract: Finite semigroups arise as syntactic semigroups of regular languages and as transition semigroups of finite automata. Among the most important and intensively studied classes of finite semigroups are the partial transformation semigroup and the semigroup of all order-preserving partial transformations. The purpose of this paper is to study a new semigroup of strongly orderpreserving partial transformations in the view of semiring theory.

A Proper Cover on a Right Type B Semigroup

Abstract: The concept of a proper cover of a right type B semigroup is introduced.Furthermore,it is proved that any proper cover for a right type B semigroup is a proper cover over a left cancellative monoid.A structure theorem of proper covers for a right type B semigroup is also given.

Ascending chain conditions on principal left and right ideals for semidirect products of semigroups

Abstract: In this paper we investigate the ascending chain conditions on principal left and right ideals for semidirect products of semigroups and show how this is connected to the corresponding problem for rings of skew generalized power series. Let S be a left cancellative semigroup with a unique idempotent e, T a right cancellative semigroup with an idempotent f and \(\omega: T \to \operatorname {End}(S)\) a semigroup homomorphism such that ω(f)=idS. We show that in this case the semidirect product S⋊ωT satisfies the ascending chain condition for principal left ideals (resp. right ideals) if and only if S and T satisfy the ascending chain condition for principal left ideals (resp. right ideals and \(\operatorname {Im}\omega(t)\) is closed for complete inverses for all t∈T). We also give several examples to show that for more general semigroups these implications may not hold.

Semigroup actions on posets and preimage quasi-orders

Abstract: Structures consisting of a semigroup of (partial) functions on a set X, a poset of subsets of X, and a preimage operation linking the two, arise commonly throughout mathematics. The poset may be equipped with one or more set operations, up to Boolean algebra structure. Such structures are finitely axiomatized here in terms of order-preserving semigroup actions on posets. This generalises Schein’s axiomatization of semigroups of partial functions equipped with the first projection quasi-order.

Classification of finite commutative semigroups for which the inverse monoid of local automorphisms is permutable

Abstract: We present a classification of finite commutative semigroups for which the inverse monoid of local automorphisms is permutable.

A cauchy-davenport theorem for semigroups

Abstract: We generalize the Davenport transform and use it to prove that, for a (possibly non-commutative) cancellative semigroup A =( A, +) and non- empty subsets X, Y of A such that the subsemigroup generated by Y is commu- tative, we have |X + Y |≥ min(ω(Y ), |X| + |Y |− 1), where ω(Y ): = sup y0∈Y ∩A× inf y∈Y \{y0 } |� y − y 0�| . This carries over the Cauchy-Davenport theorem to the broader setting of semi- groups, and it implies, on the one hand, a common extension of I. Chowla's and S.S. Pillai's theorems for cyclic groups, and on the other a significant strength- ening of another generalization of the same Cauchy-Davenport theorem (to com- mutative groups), where ω(Y ) in the above is replaced by the infimum of |S| as S ranges over the non-trivial subgroups of A.

Structure theorems for weakly B-abundant semigroups

Abstract: The aim of this article is to provide structure theorems for weakly B-abundant semigroups satisfying the Congruence Condition (C), where B is a band. Such semigroups may be thought of as generalisations of orthodox semigroups. Our focus is on providing a description of a weakly B-abundant semigroup S with (C) as a spined product of a weakly B-abundant semigroup S B (depending only on B) and S/γ B , where γ B is the analogue of the least inverse congruence on an orthodox semigroup. This result is an analogue of the Hall-Yamada theorem for orthodox semigroups. In the case that B is a normal band, or S is weakly B-superabundant, we find a closed form for γ B . In addition, we build on an existing result of Ren to show that a weakly B-superabundant semigroup (C) is a semilattice of rectangular bands of monoids.

Endomorphism semigroups of some free products

Abstract: We define the notion of a w-class of semigroups and prove that every endomorphism semigroup of a free product of semigroups from a maximal w-class is isomorphic to a wreath product of a transformation semigroup with some small category.